On Countable Compactness and Sequential Compactness

Abstract

If a countably compact 73 space X can be expressed as a union of less then c many first countable subspaces, then MA implies that X is sequentially compact. Also MA implies that every countably compact space of size < c is sequentially compact. However, there is a model of ZFC in which u,<c and there is a countably compact, separable 72 space of size u,, which is not sequentially compact. It is well known that every sequentially compact space is countably compact, but the reverse is false. Even compact spaces need not be sequentially compact. It is interesting to note that by adding some restrictions on the size of spaces, (countable) compactness of spaces is sometimes enough to guarantee sequential compactness. For instance, any compact space of size w, is sequentially compact (L) (also see (F, MS and W)). natural question is whether countably compact spaces of size < c are sequen- tially compact. The main result of this paper shows that it is undecidable in ZFC. w* means su\u, and for any P G u, P* = (C\su P)\P is a clopen subset of u*. Let A be a space with a dense subset = {ai: i < u}. Each x G X corresponds to a closed subset of «*; namely, Cx = H {{z < w: ai G U}*: U is a neighborhood of x}. Let t/* = {z: a, G U}*.